Glass pane not acting the way it should
Your 'Fake Glass' using a single plane instead of a solid works as expected when the plane is viewed from the 'front' side (the side that the normal is facing) but the Fresnel node produces different results when viewed from the 'back'. In order to correct for this you need to adjust the IOR to 1/IOR when viewing the surface from the 'back' and this can be achieved as follows :



Here the Backfacing geometry node controls the Mix between IOR and 1/IOR so that when viewed from the 'back' of the surface it still behaves the same as from the 'front'.

Replace the Fresnel node in your example with the above nodes and this should resolve your problem.